Finite Group Databasebeta

← #29707493

Subgroups — #29707493

Order 15616.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C2×#16839800C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic, complemented
4C22×#14519689C211yesmaximal, characteristic, complemented
5#21448198C211yesmaximal, characteristic, complemented
6#21447021C212yesmaximal, complemented
7#21447014C212yesmaximal, complemented
8#21447007C212yesmaximal, complemented
9#21447000C212yesmaximal, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C2×C61×#416C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840269—24no—
15#16840269—24no—
16C2×#14519758—21no—
17C2×#14519753C2212yes—
18C2×#14519748C2212yes—
19C2×#14519745—21no—
20C2×C4×C22⋊D61—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839800—24no—
24C2×#14519711C2212yescomplemented
25C2×#14519709C2212yescomplemented
26#16839758—24no—
27C2×#14519701C2212yescomplemented
28C2×#14519698C2212yescomplemented
29C2×#14519689—24no—
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839623—21no—
33#16839619C2212yes—
34#16839619C2212yes—
35#16839617—21no—
36C4×#11758198—22no—
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39C22×#41×C61C2211yescharacteristic
40C23×C8×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#416—24no—
43C2×#103×C61C2211yescharacteristic, complemented
44C2×#103×C61C2211yescharacteristic, complemented
45C2×#102×C61C2211yescharacteristic, complemented
46C2×#102×C61C2211yescharacteristic, complemented
47C24×D61D411yescharacteristic
48C22×C22⋊D61—22no—
49C23×Dic61D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61D412yes—
57C22×C4×D61C2312yes—
58C22×#92418—22no—
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418C2312yes—
62#14519758—28no—
63#14519753—24no—
64#14519753—24no—
65#14519748—24no—
66#14519748—24no—
67#14519745—28no—
68C4×C22⋊D61—216no—
69#14519736—22no—
70#14519736—22no—
71#14519736—22no—
72#14519736D414yes—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84#14519711—24no—
85#14519709—24no—
86#14519701—24no—
87#14519698—24no—
88#14519689—24no—
89#14519689—24no—
90#14519684—22no—
91#14519684—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61D414yes—
98C22×Q8×C61C2311yescharacteristic
99C22×D4×C61—21no—
100C22×D4×C61C2311yescharacteristic
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C22×C8×C61—22no—
104C22×C8×C61—22no—
105C22×C8×C61D414yescomplemented
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61—22no—
109#117×C61—22no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—22no—
115#103×C61—22no—
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C23×D61—22no—
121C23×D61—24no—
122C23×D61—22no—
123C23×D61—22no—
124C23×D61—42no—
125C23×D61C2×D412yes—
126#41×C61—28no—
127#41×C61—28no—
128#41×C61—44no—
129#41×C61—44no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61#4112yes—
156C22×C4×C61#4112yes—
157C22×C4×C61#4112yes—
158C22×C4×C61C2411yescharacteristic
159C2×C22⋊D61—28no—
160C2×C22⋊D61—44no—
161C2×C22⋊D61—44no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61C2×D412yes—
168C2×C4×D61—24no—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—44no—
178C2×C4×D61—44no—
179C2×C4×D61—44no—
180C2×C4×D61—44no—
181C2×C8×C61—28no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C2×D412yes—
187C2×C8×C61C2×D412yes—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C22⋊C4×C61—42no—
193C42×C61—42no—
194C22⋊Dic61—28no—
195C22⋊Dic61—24no—
196C22⋊Dic61—42no—
197#11758198—28no—
198#11758198—44no—
199#11758198—44no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C2×D414yes—
204#11758196—28no—
205C4×Dic61—24no—
206C4×Dic61—42no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—44no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61#12012yes—
219C23×C61#12112yes—
220C23×C61#12312yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—28no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—48no—
230C22×D61—48no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61#13514yes—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—48no—
241D4×C61—41no—
242D4×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—48no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61#12012yes—
252C2×C4×C61#12114yes—
253C2×C4×C61#12114yes—
254C2×C4×C61#12112yes—
255C2×C4×C61#12314yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12414yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C22⋊D61—416no—
265C2×Dic61—24no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—44no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C4×D61—42no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—48no—
280C8×C61—22no—
281C8×C61—22no—
282#92418—42no—
283#23479—611nomaximal, Sylow
284C22×C61—24no—
285C22×C61—24no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—42no—
289C22×C61—44no—
290C22×C61—82no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61#39414yes—
294C22×C61C2×#12011yescharacteristic
295C22×C61C2×#12111yescharacteristic
296C22×C61C2×#12311yescharacteristic
297C22×C61C2×#13512yes—
298C22×C61#57512yes—
299C2×D61—24no—
300C2×D61—24no—
301C2×D61—24no—
302C2×D61—416no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C4×C61—42no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61#39414yes—
311C4×C61C2×#12111yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12311yescharacteristic
314C4×C61C2×#12411yescharacteristic
315Dic61—42no—
316Dic61—44no—
317Dic61—82no—
318C2×#525—611no—
319C2×#427—611no—
320C2×#416—611no—
321C2×#416D6111yescharacteristic, complemented
322C22×C22⋊C8—611no—
323#2280—611no—
324#2036—612no—
325#2035—612no—
326#2034—612no—
327#2033—612no—
328#1735—611no—
329C2×C61—24no—
330C2×C61—82no—
331C2×C61C2×#39411yescharacteristic
332C2×C61#300212yes—
333C2×C61#305011yescharacteristic
334C2×C61#305411yescharacteristic
335C2×C61#309112yes—
336D61—44no—
337C22×#41C2×D6111yescharacteristic
338C24×C4—611no—
339C23×C8—611no—
340C23×C8C2×D6111yescharacteristic
341#525—1224no—
342#525—1224no—
343C2×#123—1221no—
344C2×C22⋊Q8—612no—
345C2×#121—612no—
346C2×#120—1221no—
347C2×C4×D4—1222no—
348C2×#117—611no—
349C2×#117—611no—
350C2×#117C2×D6111yescharacteristic
351#427—1224no—
352C2×#107—612no—
353C2×#106—612no—
354#416—24no—
355#416—1224no—
356C2×#103—612no—
357C2×#103C2×D6111yescharacteristic, complemented
358C2×#103C2×D6111yescharacteristic, complemented
359C2×#102—612no—
360C2×#102C2×D6111yescharacteristic, complemented
361C2×#102C2×D6111yescharacteristic, complemented
362C2×C22⋊C8—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8—1224no—
365#381—1221no—
366#380—612no—
367#380—612no—
368#379—1221no—
369C4×C22⋊C4—1222no—
370#340—612no—
371#340—612no—
372C61#2347911yescharacteristic, complemented, Sylow
373C25—611no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41D24414yes—
380C22×Q8C22×D6111yescharacteristic
381C22×D4—21no—
382C22×D4—1222no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—21no—
385C23×C4—611no—
386C23×C4—612no—
387C23×C4—612no—
388C23×C4—1222no—
389C23×C4—1222no—
390C23×C4—1222no—
391C23×C4—1222no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4C22×D6111yescharacteristic
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—612no—
398C22×C8—1222no—
399C22×C8—1222no—
400C22×C8—1222no—
401C22×C8C22⋊D6114yescomplemented
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404#123—1228no—
405C22⋊Q8—1224no—
406C22⋊Q8—1224no—
407#121—1224no—
408#121—1224no—
409#120—1228no—
410C4×D4—12216no—
411#117—22no—
412#117—22no—
413#117—614no—
414#117—1222no—
415#117—1222no—
416#117—1222no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1221no—
429C2×C42—21no—
430C2×C42—1221no—
431C2×C42—1221no—
432#107—1224no—
433#106—1224no—
434#103—22no—
435#103—22no—
436#103—1224no—
437#102—22no—
438#102—22no—
439#102—1224no—
440C22⋊C8—1224no—
441C22⋊C8—1224no—
442#95—1222no—
443#95—1222no—
444C24—21no—
445C24—612no—
446C24—1224no—
447C24—1222no—
448C24—1222no—
449C24—1222no—
450C24—2442no—
451C24D4×D6111yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×D24412yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—1228no—
471C2×D4—2444no—
472C2×D4—2444no—
473C2×D4C2×D24412yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—612no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1224no—
494C22×C4—1222no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4—2444no—
498C22×C4—2444no—
499C22×C4C2×D24412yes—
500C22×C4#1175822212yes—
501C22×C4D4×D6112yes—
502C22×C4D4×D6111yescharacteristic
503C22×C4#1175822412yes—
504C22×C4#1175822412yes—
505C22×C4C2×C22⋊D6112yes—
506C22×C4C23×D6111yescharacteristic
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—28no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C2×C8—2444no—
516C2×C8C2×C22⋊D6112yes—
517C2×C8C2×C22⋊D6112yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—614no—
523C4⋊C4—1228no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C22⋊C4—42no—
528C22⋊C4—1228no—
529C22⋊C4—1228no—
530C22⋊C4—1224no—
531C22⋊C4—2444no—
532C22⋊C4—2444no—
533C22⋊C4—2442no—
534C42—42no—
535C42—1224no—
536C42—2442no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—614no—
545C23—1228no—
546C23—1222no—
547C23—1222no—
548C23—1222no—
549C23—1222no—
550C23—1222no—
551C23—1222no—
552C23—2448no—
553C23—2448no—
554C23—2444no—
555C23—2444no—
556C23#1451974712yes—
557C23#1451974812yes—
558C23#1451975912yes—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—24416no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2444no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2448no—
593C2×C4#1451974712yes—
594C2×C4#1451974812yes—
595C2×C4#1451974914yes—
596C2×C4#1451975114yes—
597C2×C4#1451975912yes—
598C2×C4#1451976114yes—
599C2×C4#1451976314yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4C22×D24411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822411yescharacteristic
605C2×C4C2×#1175822211yescharacteristic
606C8—22no—
607C8—22no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—44no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—1224no—
619C22—1224no—
620C22—1224no—
621C22—24416no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22#1683967814yes—
627C22C2×#1451974711yescharacteristic
628C22C2×#1451974811yescharacteristic
629C22C2×#1451975911yescharacteristic
630C22#135×D6112yes—
631C22#1684055912yes—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—4882no—
636C4—2442no—
637C4—2444no—
638C4#1683967814yes—
639C4C2×#1451974911yescharacteristic
640C4C2×#1451975111yescharacteristic
641C4C2×#1451976111yescharacteristic
642C4C2×#1451976311yescharacteristic
643C2—24no—
644C2—82no—
645C2—2444no—
646C2C2×#1683967811yescharacteristic
647C2#2145503912yes—
648C2#2145539611yescharacteristic
649C2#2145542411yescharacteristic
650C2#2145573012yes—
subgroupG/H
centerC22×C4D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840269C2
hypercenterC2×#416D61
hypocenterC61#23479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839800C2semidirect
3. C2×#16839758C2semidirect
4. C22×#14519689C2semidirect
5. #21448198C2semidirect
6. #21447021C2semidirect
7. #21447014C2semidirect
8. #21447007C2semidirect
9. #21447000C2semidirect
10. #21445535C2semidirect
11. C2×C61×#416C2semidirect
24. C2×#14519711C22semidirect
25. C2×#14519709C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
43. C2×#103×C61C22semidirect
44. C2×#103×C61C22semidirect
45. C2×#102×C61C22semidirect
46. C2×#102×C61C22semidirect
105. C22×C8×C61D4semidirect
321. C2×#416D61semidirect
357. C2×#103C2×D61semidirect
358. C2×#103C2×D61semidirect
360. C2×#102C2×D61semidirect
361. C2×#102C2×D61semidirect
372. C61#23479semidirect
401. C22×C8C22⋊D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168402692
C2×#168398002
C2×#168397582
C22×#145196892
#214481982
#214470212
#214470142
#214470072
#214470002
#214455352
C2×C61×#4162
#2347961